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Step * of Lemma free-dlwc-1

∀[T:Type]
  ∀eq:EqDecider(T). ∀Cs:T ⟶ fset(fset(T)). ∀x:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])).
    (x = 1 ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])) ⇐⇒ {} ∈ x)
BY
{ ((UnivCD THENA Auto)
   THEN (Subst' 1 ~ {{}} 0 THENA (RW (SubC (SymbCompC [] 100)) 0 THEN Auto))
   THEN (All (RWO "free-dlwc-point") THENA Auto)
   THEN Auto
   THEN Try ((HypSubst' (-1) 0 THEN Auto))) }

1
1. T : Type
2. eq : EqDecider(T)
3. Cs : T ⟶ fset(fset(T))
4. x : {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))} 
5. {} ∈ x
⊢ x = {{}} ∈ {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))} 

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Latex:

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Latex:
\mforall{}[T:Type]
    \mforall{}eq:EqDecider(T).  \mforall{}Cs:T  {}\mrightarrow{}  fset(fset(T)).
    \mforall{}x:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])).
        (x  =  1  \mLeftarrow{}{}\mRightarrow{}  \{\}  \mmember{}  x)


By

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Latex:
((UnivCD  THENA  Auto)
  THEN  (Subst'  1  \msim{}  \{\{\}\}  0  THENA  (RW  (SubC  (SymbCompC  []  100))  0  THEN  Auto))
  THEN  (All  (RWO  "free-dlwc-point")  THENA  Auto)
  THEN  Auto
  THEN  Try  ((HypSubst'  (-1)  0  THEN  Auto)))

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